Concept of Symmetry in Closure Spaces as a Tool for Naturalization of Information (Algebraic system, Logic, Language and Computer Science)

Concept

of

_Symmetry

Closure

_Spaces

for

Naturalization of Information

Marcin J. Schroeder

Akita Intemational

_University,

mjs@aiu.ac.jp

1. Introduction

Tonaturalize_information,i.e.tomake the_conceptof informationasubjectof thestudyofreality

independent from _{subjective judgment requires methodology} of its _study consistent with that of

natural sciences.Section 2 of this_{paperpresents}historical_{argumentation}for the fundamental role of the_studyof_symmetriesinnaturalsciences and in the consequenceof the claim that naturalization of

information_requiresa_methodologyof the _studyofits _symmetries. Section3 includes mathematical

preliminariesnecessaryfor thestudyofsymmetriesinanarbitraryclosure space carriedoutin Section 4. Section5demonstrates theconnection of the_concept ofinformation and closure_spaces. Thissets foundations forfurtherstudies of_symmetriesof information.

2.

_Symmetry

_Methodology

Typicalaccountof the_originsof the modern_theoryof_symmetrystartswith_{Erlangen Program}of

Felix Klein_publishedin 1872. _[1] This_publicationwasofimmense influence. Klein_proposeda new

paradigm of mathematical _{study focusing} not on its objects, but on their transformations. His

mathematical_{theoryofgeometricsymmetry}wasunderstoodas aninvestigationof the invariance with

respect to transformations of the _{geometric space (two‐dimensional plane} or _higher dimensional

space).Klein used this_{verygeneralconcept}of_{geometricsymmetryfor the purpose}ofaclassification

of different _types of _{geometries (Euclidean} and _{non‐Euclidean).} The fundamental _conceptual

framework of Klein’s_Program(asitbecame functionasaparadigm)wasbasedonthe schemeof(1)

space as a collection ofpoints \rightarrow (2) algebraic structure _(group) of its transformations \rightarrow ₍₃₎

invariants ofthetransformations,i.e. _{configurations}of_pointswhich donot_changeas awhole,while

their_points are _{permuted by} transformations. Selections of_algebraic substructures _(subgroups) of

transformations _correspond to different _types and levels of invariant _{configurations allowing} to differentiate andto_comparestructural_propertiesassociated with_symmetry.The classical_exampleof the mirror _{symmetry (symmetry} with _respectto the surface of the _mirror) can be identified with

invariance with_respecttomirror reflection.

Klein’s workwasutilizinga newtheoryof_{groups which in the works}ofArthur_{Cayley [2],}Camille

Jordan _[3] and others foundits _identity as a_partofalgebra. Klein’s_{Erlangen Program} to _classify

geometries has been extended to _many other _disciplines of mathematics becoming one ofmost

common_paradigmsofthemathematical research.Also,in the consequence ofoneofmost_important

contributionstomathematical _physics of all times_publishedin 1918_{Uy Emmy}Noether _{[4] stating} that _every differentiable _symmetry of the action of a physical _system has a corresponding

conservation_law,the invariance with_respectto_{transformations,}i.e._symmehywith_respecttothese transformations, became the central _subjectof_physics. The fact that the conservation laws for the

physical magnitudes suchas _{energy,momentum,} angularmomentumareassociatedUythe theorem

with transformations _{describing changes} of reference _frames, i.e. observers makes the study of symmetryacentral tool for scientific_{methodology. Physics (and}science in_general)is_lookingforan

objective descriptionof_reality,i.e._descriptionthat is invariantorcovariant with_changesof observers. Noether’s theoremtellsusthat suchdescriptioncanbe carriedoutwith conserved_magnitudes.

The year 1872 whenErlangen Program waspublishedcanbe considered the_{starting point}of the studyof_symmetriesintermsof the grouptheory,butnotof thescientific_studyof_symmetries. The

importanceof_symmetries,stillunderstood intermsofthe invariance but ofthe mirror reflections_only

wasrecognized longbeforeErlangen Programinthecontext_{ofbiochemistry.}LouisPasteur_published

in 1848 one ofhis most _{important papers explaining}isomerism oftartrates, more _specificallyof

tartaric acid_bythemolecular_{chirality.[5]} Heshowedthat the differences between_{optical properties} of the solutions of this _{organic compound} between _{samples synthesized} in _{living organisms} and samples synthesized artificiallyresult from the fact that in_{artificially synthesized}molecules_although constructed from the same atoms as those in natural _synthesis, have two_{geometric configurations}

whichare_symmetricwith_respecttothemirrorreflection,butnot_{exchangeable Uy spatial}translations

orrotations (thesame _way as left and right palms of human_hands), while in the nature _onlyleft‐ handed_{configurations}occur. Laterit tumedoutthatalmost_{exclusively naturally synthesized}amino acids_(andtherefore_proteins)are_{“left‐handed”,}andsugarsare_“righthanded”.Artificial_synthesis,if not constrained _{by special procedures} leads to _{equal production} of the left and _right handedness. There isno_{commonly accepted explanation}ofthis_{mysterious phenomenon}eventoday.

Thestructural characteristic which_givesthe distinctionofleft and_righthandednesswasgivenname

of_chirality. Thus, our hands are _chiral, while _majority of_{simple organisms} are symmetric with

respectto _rotations, and therefore _{achiral). Chirality} of molecules became one ofmost _important

subjectsofthe 19^{\mathrm{t}\mathrm{h}}_{Century biochemistry leading}tothe_discoveryofthe role oftheatomsof carbonin

formation of chiral molecules formulated into Le Bel —van’t Hoff Rule _{published by} these two researchers_{independently}in1874.

The_studyof_symmetryin_biology,in_particularof_chiralityin_{complex organisms} couldnot have

been _explained in the 19^{\mathrm{t}\mathrm{h}} Century, but researchers _published some phenomenological laws of

evolutionand_{phenotypic development}of_organisms, suchasBateson’s Rule. Much later Bateson’s

sonprovided explanationof this rulein termsofinformation science. _{[6, 7]}Similar_{interpretation}can

be _givento Curie’s _{Dissymmetry Principle.}Pierre Curie made so _{manyimportant} contributionsto

physicsand_chemistrythatthis fundamental_principleof_{greatimportance} is_rarely invoked.Its little bit outdated_originalformulation_usingtheterm_{“dissymmetry”}insteadofnow common“asymmetry”

was:Aphysicaleffectcanmothaveadissymmetryabsent from itsefficientcause.

The real _importance of these _{early developments} could be _{fully appreciated} ahalf_century later

whenit became_fullyclear thankstoadvances in_{physics(elementaryparticletheory)}thatthe_studyof the conditions for_{maintainingsymmetry}isno moreimportantthan thestudyofbreaking_symmetry.

By the mid‐20th _Century the _study of_symmetry became a fundamental tool for mathematics,

physics,chemistryandfor several branches of_biology.Thiscan_explainsudden_explosionof interest

in_{symmetryamongphilosophers.}The _swingof the_pendulum of_{dominating philosophical} interests

between _seeking an objective methodology for philosophical inquiry inspired Uy scientific

methodologyand _{introspective} andtherefore _{subjective phenomenal experience}reached the side of the former.

The_beginningsof structuralism canbetracedtotheworks of Ferdinand deSaussureon_linguistics

(more specificallyhis lectures 1907‐1911 _{posthumously published by}his _disciplesin 1916_[8]).The emphasis onthe structural characteristics of_language andon their synchronous analysis prompted

increased interest in the _meaningof the _conceptofstructure. Itwas anatural _consequencethat the

toolsusedinscience for the structural_analysisintermsof_symmetryfound their_wayto_psychology, anthropology and _philosophy. The most clear_programmatic work on structuralism _by Jean _Piaget publishedoriginallyin 1968is_{referringexplicitly}tothe_conceptof the_groupoftransformations. [9]

Piagetbased his_theoryof child_developmentonthesocalled Klein’s_(sic!)“four_group”.The works ofothers, for instance of Claude Levi‐Strauss, also _{employed directly} the methods _developed in consequenceofErlangen Program. [10]

The_swingof the_pendulumreversed its directionandstructuralism lost his_{dominating position}to

its_critique,but its_importancecanbeseenin thenameof this reversed_swingas_{“post‐structuralism”.}

Some ofthis criticism is naive. For instance, structuralism was criticized as _“dry” or “too much

formal”. The definite record of_“dryness” is held and _{probably always} will Ue held _by Aristotle,

togetherwith the titleof themostinfluential_philosopherof all times. More_{justified objection}of the lack of_explanationofthe_originofthestructuresconsideredinthe studies of_Piaget,Levi‐Strauss and

others and the_{missing evolutionary}or_{dynamic theory}ofstructurescanbe blamedontheseauthors, but it is more a matter of _{misunderstanding} of the mathematical tools. _{Physics, chemistry} have

powerful dynamictheories oftheir structures,sothere isnogoodreasontobelieve that such_dynamic approachis_impossiblein_philosophy.

Symmetry canbe easilyidentified inthe studies ofvisual artsand music. _Actually,the structural study of music initiated _{by Pythagoreans found its way}to mediaeval _philosophy via_Neoplatonic

authorsand thentotheworks ofthe foundersofmodernscience suchasJohannesKepler.The music

ofHeavens,understood_literallyasmusicproduced bythe motion ofthe_planetswasamathematical

model of the universe. In modern mathematics we can easily understand the reasons for the

effectiveness of such models. Modem _{spectral analysis} in _physics is not _{very far from the}

decompositionof functions_{describing physical phenomena}intohamonic_components,thesame_way as_recordingof music in_digitalformat is done.

Theuniversal characterof the_studyof_symmetryinthe_spiritof Klein’s_{Erlangen Program}became

commonly recognized inthe second half of the 20^{\mathrm{t}\mathrm{h}}_Century. The immense_popularityof the book

“Symmetry” byHermann_{Weyl greatly}contributedtothis_{recognition.[11]}At_{this time grouptheory}

in thecontext of_symmetriesbecame aneveryday tool for allphysicists andassumed a_permanent

placein_universitycurricula forstudies in_{physics, chemistry}and_{biology. [12]}Thestatement froman

article _published in Science in 1972 _by a future Nobel Prize laureate in _{Physics Philip} Warren

Anderson“Itis_{onlyslightlyoverstating}thecaseto_saythat_physicsisthe_studyof_symmetry”was alreadyatthattime_{commonly accepted}truth._[13]

Study of_symmetry became afundamental _{methodological} tool. Anderson’s article wasnot _only closing the _century of the _development of this tool, but it also included _{another very important}

message.Andersonemphasizedthe role of“breaking symmetry”. Hedemonstrated that the_physical

realityhas ahierarchic structure of_{increasingcomplexity} and that thetransition fromone level of

complexitytothe nextis associated with_{breakingsymmetry}understoodasatransformation of the

group ofsymmetryto another oflower level.Thus, not_onlythe_studyof_symmetry,but also of the

ways of itschangesisimportant.For this purpose theconceptof_symmetryhastohave_{very clear and}

preciseformulation._{Unfortunately}thereare_manycommonmisunderstandings.

Themost_{typical misunderstanding}isa_consequenceofmisinterpretationof Klein’s_Program.The

missingpartis the role of_{projectivegeometry.}Klein didnotconsider_arbitrarytransformations ofthe plane (orsetof_pointsonwhich_geometryis defined),butonlythose whichare_preservingthismost fundamental _geometric structure. This _{very important, but very frequently ignored aspect} of the

ProgramwasclearlydescribedinWeyl’sbook_{popularizingsymmetry}inthe_generalaudience: “What has all thistodo with_symmetry?It_providesthe_adequatemathematical_languagetodefine it. Givena

spatial configurations^{\infty},thoseautomorphismsofspace which leave s\leftarrowunchangedformagroup $\Gamma$,and

this _{group describes exactly} the _{symmetrypossessed by} s^{\infty}. Space itself has the full symmetry

correspondingtothe_{group of all automorphisms,}of all similarities. Thesymmetryofany figurein space is describedbyasubgroupofthat_group [11]

The _{methodological aspects} of the _study of_symmetry in _{physics suggest} that the _concept of

informationcanbenaturalized,i.e.canbecomea_partof the scientificdescriptionofreality,ifwecan develop methods of _study of information in terms of _symmetry. But to _develop a theory of

information _symmetry we have to _generalize the _concept of_symmetry from the _{closure space}

3.

_Algebraic

The_followingnotation and_{terminological}conventions will beused_throughoutthetext:

Fin(S) is a set of all finite subsets of the setS. Greekletters such as _{$\phi$, $\varphi$, $\Theta$}, etc. will indicate

functionsonthe elements ofa_givensetand withthe values _belongingto aset. Small Latin letters suchas\mathrm{f},_\mathrm{g}, \mathrm{h},etc. will indicatefunctions definedonthe subsets ofagivensetand with the values

whicharesubsets ofthisset.The doubleuseofthesymbol

$\varphi$^{-1}(\mathrm{A})

function of $\varphi$, and as aninverseimageofasetA withrespecttofunction $\varphi$ which does nothave

inverse,shouldnotcauseproblems.Thecompositionoffunctions will be writtenas_{ajuxtaposition}of

their_symbols,unlessthe fact oftheuseofacompositionof functions is contrasted withconstructing

function_images.The_symbol\congindicatesa_{bijective correspondence}orisomorphism.Throughoutthe

paper,partiallyorderedsetsareoftencalled_posets.

Preliminariesinclude several_propositionswithout_proofs, some_belongtothefolkloreof the subjectandarewell_known, somehave_{proofs straightforward.} Anintroductiontothe_subjectcanbe

found in Birkhoffs “LatticeTheory”. [14]

DEFINITION 3.1 Let _fbe a_{function from} the_powerset _ofaset Sto _itselfwhich_satisfiesthe followingtwoconditions:

(l) VA\subseteq S:A\subseteq f(A),

(2) VA,B\underline{c}S.\cdot A\subseteq B\Rightarrow f(A)\subseteq f(B), (3)VA\subseteq S.\cdot ff(A)=f(f(A))=f(A).

Then_fiscalledan_operator(ortransitiveclosure_operator)onS. Theset_ofalloperatorsontheset

S isindicated_{by I(S).}Aset_equippedwithaclosure_operatorwillbe calledaclosure space_<S,f>.

The third conditionscanbereplaced byacondition: whichis easiertouseinproofs, but whichin

combination with othertWo_givesexactlythesame_concept:

(3^{*})VA_{2}B\subseteq S.\cdot A\subseteq f(B)\text{ニ}>f(A)\subseteq f(B).

The strongerform ofthis condition_{VA,B\subseteq S.\cdot A\subseteq f(B) iff f(A)\subseteq f(B)}canbe used insteadofallthree

conditions to _define a transitive _{operator, but this fact} does not have a _{significant practical}

importance.

DEFINITION 3.2 _Letfbeaclosure_operatoron asetS. The subsets A_{ofS satisfying}thecondition

f(A)=A, calledf‐closedsetsform a Moorefamily f‐Cl, i.e. it is closed with respectto arbitrary

intersections andincludes thesetS(whichcanbe consideredthe intersection_oftheemptysubfamily of subsets). EveryMoore_{family M defines} a transitive _{operatorf(A)=\displaystyle \cap\oint M\in M:A\subseteq MJ}. Set

theoreticalinclusion_definesapartialorderon f‐Clwith_respecttowhich it isacompletelattice. To

thisstructurewewillreferasthe_completelattice_{L_{f} off‐closed (orjust closed)}subsets.

Let_{fand g beoperators}on asetS. Therelation_{defined byf\underline{<}g if VA\subseteq S:f(A)\subseteq g(A)is}apartial

orderon I(S), withrespect towhich it is acomplete lattice. Thispartial ordercorresponds tothe

inverse_oftheinclusion_ofthe Moore_{families ofclosedsubsets}

DEFINITION 3.3_Letfbeaclosure_operatoron asetS,gaclosure_operatoronsetT,and $\varphi$ bea

function from Sto T. The_{function $\varphi$ is ffg)‐continuous if VA\subseteq S.\cdot $\varphi$ f(A)\underline{c}g $\varphi$(A)}. We willwrite

continuous,_{ifno confusion}is_likely.

PROPOSITION 3.1 _{Continuity of}the_{function $\varphi$} as definedabove is _equivalent to each _ofthe

followingstatements:

(2)

VB\subseteq T.\cdot f$\varphi$^{1}(B)-\subseteq$\varphi$^{\mathrm{J}}g(B)-,

VB\underline{c}T:ff$\varphi$^{1}(B)\subseteq g(B)-.

VB\in g-C.\cdot $\varphi$(B)\in f-C-.

DEFINITION 3.4 _Letfbeaclosure_operatoron asetS,gaclosure_operatoronsetT,and $\varphi$ bea

functionfrom StoT. The_function $\varphi$ is (fg)‐isomorphism ifit is_bijectiveand _{VA\underline{c}S:ff(A)=g $\varphi$(A)}.

Wewillwrite_{isomorphism, ifno confusion}is_{likely. IfS=T}, wewill call $\varphi$an(fg)‐outomorphism, or

smply automorphism.

PROPOSITION 3.2 Theconditions_forafunction _$\varphi$tobe an _isomorphism, as_{definedabove,} are

equivalenttoeitheronebelow:

(1) $\varphi$ hasaninverse

$\varphi$^{l}-

(2)Thereexists_{afunction $\psi$from} TtoSsuch that _{$\varphi \psi$=id_{T}} and _{$\psi \varphi$=id_{S} and both $\varphi$ and $\psi$} are

continuous.

PROPOSITION 3.3_Letfbeaclosure_operatoron aset S,gaclosure_operatoronsetT, and $\varphi$ bea

functionfrom StoT. Then, every(fg)‐isomorphism $\varphi$generatesalattice isomorphism$\varphi$^{*}between the

completelattices_ofclosed subsets_{L_{f}}and_{L_{g} defmed byVA\in L_{f}\cdot$\varphi$^{*}(A)= $\varphi$(A)\in L_{g}}.Also, ifafunction

$\varphi$:S\rightarrow Tis_bijectiveandis_generatingalatticeisomorphism $\varphi$^{*}betweenlatHces L and_Lg., then _{$\varphi$ is} an rgJ‐isomorphism.

COROLLARY 3.4_{Everyf‐authomorphism $\varphi$ of<SJ>generates}a_uniquelattice_{automorphism of}

L However, more than onef‐authomorphism _$\varphi$ of<Sf>can correspond to the same lattice

automorphismofL_{f}

PROPOSITION 3.4 The set _ofall_{f‐automorphisms of<Sf> forms}a_groupAut<SJ> under the

function composition. This_{group isisomorphic}to_Aut(Ldoflattice_{automorphisms ofL}

Wewillrefertothe_conceptofan_(antisotone)Galois connectionbetweentwo_posets.

DEFINITION 3.5Let<P,\underline{<}>and<Q,\underline{<}>be posets and $\varphi$ and $\psi$ be anti‐isotone(order inverting) functions $\varphi$:P\rightarrow Qand _{$\psi$ Q\rightarrow P}.Then thefunctions defineaGalois connectionbetween the posets

if. \mathrm{t}\mathrm{l}\mathrm{X} $\epsilon$ P:x\underline{<} $\psi \varphi$(x)and_{Vy $\epsilon$ Q.\cdot y\underline{<} $\varphi \psi$(y)}.

Galois connectioncanbedefinedinanequivalent_wayas a_pairoffunctions $\varphi$:P\rightarrow Qand _{$\psi$ Q\rightarrow P}

suchthat _{Vx\in PVy\in Q.\cdot y\underline{<} $\varphi$(x)iff x $\Xi \psi$(y)}.

PROPOSITION 3.5_Ifa_pairoffunctions $\varphi$:P\rightarrow Qand _{$\psi$\cdot Q\rightarrow P}definesaGalois connection, then

the_{functions $\psi \varphi$.\cdot P\rightarrow P}and _{$\varphi \psi$\cdot Q\rightarrow Q}areclosure operators, i.e. they satisfythe conditions1)-3) of

Definition3.1_{generalizedfrom}theinclusion \underline{\subseteq}to thepartialorder‐<.Moreover,thefunctions $\varphi$.\cdot P\rightarrow

Qand _{$\psi$ Q\rightarrow Pdefine}order_{anti‐isomorphism (orderreversingfunctionspreserving}all_infimaand

suprema) betweenthe_completelattices_ofclosedelements inthe_posetsPand_Q.

PROPOSITION 3.6 Given an anti‐isotone function $\varphi$:P\rightarrow Q. If thefunction $\varphi$.\cdot P\rightarrow Q defines

togetherwith _{$\psi$ Q\rightarrow P}aGalois connection,then thefunction _{$\psi$ is unique. However,} thereareanti‐

isotone_functionswhich do_notformaGaloisconnectionwith_anyfunction.

PEOPOSITION 3.7_Ifposets<P,\underline{<}>and_<Q,\underline{<}>are_{completelattices,}then_forevery anti‐isotone function $\varphi$:P\rightarrow Q, thereexists(byProp. 3.6unique) junction $\psi$ Q\rightarrow P, such thatthey formaGalois

connection._{Thefunction $\psi$ Q\rightarrow P}is_{defined by: $\Phi$\in Q.\cdot $\psi$(y)=\vee\{x\in P:y\underline{<} $\varphi$(x)J}, whereisthe lowest

upperboundoftheset, whichmustexist inacompletelattice.

4.

_Concept

_Symmetry

_Spaces

Intheabstract formulation of_geometryontheplaneinthetermsofclosure_{spaces theonly}closed subsets are entire plane, _{empty subset, points} and _straight lines. Geometric _{configurations} are

collectionsof_points orlines. However, the_concepts ofclosure spaces do not_giveus_{any tools for}

analysisofsuch_{configurations beyond}the intersections oflines_{producing points}and_pairsof_points

defininglines. Our_goalisto _providethe tools for the_analysis of such_{configurations} not_onlyfor

abstract_geometries,but for_arbitraryclosure_spaces.The_{approach presented}belowwasinformedby

the_analogywith_{geometric symmetries}inthe choice of_grouptheoryasafoundation. Sincewewill use only rudimentary facts about _{group actions} on a _{set, there} will be no need for extensive

explanationof the_conceptsof this_theory.

We will use in the _presentation of the _approach tothe _study of_symmetry of_{configurations} a

selectedclosurespace<S,f>with thegroup\mathrm{G}=\mathrm{A}\mathrm{u}\mathrm{t}<S,j> of its \mathrm{f}_{‐automorphisms.}A_{configuration} inthisspacewill beanarbitrary,butnot_emptyset s\leftarrowof \mathrm{f}‐closed subsets ofS. Itisanatural question

how the_completelattice of_subgroupsofthe_{group \mathrm{G}}is relatedto_symmetriesof_{configurations,}i.e.to

symmetriesof subsets ofthe_completelattice\mathrm{L}_{\mathrm{f}}of closedsubsetsin_<S,f>.

We will start from a _simple observation related to the _{generalization} ofone of _examples in BirkhoflPs“Lattice_{Theory” [14].}Its_proofisso_elementarythat it is leftas anexercise.

LEMMA 4.1 Let H beasubsetofa_{group Gacting}on asetS,such that theidentity$\epsilon$_{G}ofGbelongs

toH_Definethe_family\lrcorner^{\sim_{H}}_ofsubsets_ofS_by VA\subseteq S:A\in J_{H}^{\sim}iff \mathrm{h}\in AV $\varphi$\in H: $\varphi$(x)\in A. Then s_{H}^{\leftarrow}isa

completelattice with_respecttotheorder_ofinclusion_ofsets.

Toavoid_comingto toofast conclusionwehavetonotice thatwearenotinterested in stabilizers of setsof elementsof the_{closure space<S,f>},Uut of the families of closedsubsets. Thereforewehave

to _applythis lemmato the families ofsets of closedsubsets of<S,f>. We will use the notation

introducedin the_previoussectionand the_conceptsdefined and_explainedthere.

PROPOSITION 4.2 Let H beasubgroup ofthe groupG=Aut(L).Definethefamilyd_{H}ofsubsets

ofL_{f} by VK\subseteq L_{f}\cdot K\in \mathrm{c}f_{H} iff VA\in KV $\varphi$ eH:$\varphi$^{*}(A)\in K. Thend_{H}isacompletelattice with respecttothe

order_{ofinclusion ofsets.}

PROPOSITION 4.3 Function $\Phi$:H\rightarrow d_{H}definedin_Prop.4.2 is anti‐isotone_functionbetweentwo posets, oneofthem (thelattice_{ofsubgroups ofagroup G)}isa_completelattice.

Proof: Let \mathrm{K}beasubgroup ofH.Then _{\forall \mathrm{K}\subseteq \mathrm{L}_{\mathrm{f}}:\mathrm{K}\in$\theta$_{\mathrm{H}}}iff_{\forall \mathrm{A}\in \mathrm{K}\forall $\varphi$\in \mathrm{H}:$\varphi$^{*}(\mathrm{A})\in \mathrm{K}}.But\forall $\varphi$\in \mathrm{K}:

$\varphi$\in \mathrm{H},therefore\mathrm{K}\in$\theta$_{\mathrm{K}}.

Nowwecandefine aGalois connection. _ByProposition3.7 and Remark 3.8we know that there

exists a Galois connection betweenthe _poset of_complete lattices $\theta$_{\mathrm{H}} and the complete lattice of

subgroupsof_{\mathrm{G}=\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{j}_{\llcorner \mathrm{f}})\cong}Aut_<sp.

PROPOSITION 4.4The_{following twofunctionsform}aGalois connection:

$\Phi$:H\rightarrow d_{H}_{defined by VK\subseteq L_{f}K\in d_{H} iff VA\in KV $\varphi$\in H:$\varphi$^{*}(A)\in K}and

$\Psi$: $\theta$\rightarrow H_{defined by \vee $\gamma$ K subgroup of G: $\theta$\displaystyle \subseteq $\epsilon$ f_{K}J=\int $\varphi$\in G: $\varphi$( $\theta$)\subseteq $\theta$ j}.The lastequality is a

consequenceofthefactthat\displaystyle \oint $\varphi$\in G: $\varphi$( $\theta$)\subseteq $\theta$ jisasubgroup ofG.

5.

_Symmetry

Inorderto combineboth_aspectsof information andto_placethis_conceptin thecontextofnon‐

trivial _{philosophical conceptual} framework, the _present author introduced his definition of informationintermsoftheone‐_{manycategorical opposition}witha_verylongandrich_{philosophical}

wordsasthat, which makesoneoutof_{many. There}aretwo_{ways in which many}canUe madeone,

either_bythe selection ofoneout _{of many,}orby bindingthe manyintoawhole_bysomestructure. Theformer isaselective manifestationofinformationand the latter isastructural manifestation.They

aredifferent manifestations ofthesame _conceptofinformation,notdifferent_types, asoneis_always

accompanied bytheother,althoughthe_{multiplicity (many)}canbe differentin eachcase.

This dualism between _coexisting manifestations was _{explained by} the author in his earlier expositionsofthe definition_usingasimple exampleof thecollectionofthe_keystoroomsinahotel. It is _easy to _{agree that} theuse of_keys is based on theirinformational _{content, but information is}

involved in this use in two different ways, _through the selection of the _rightkey, orthrough the

geometric descriptionof its_shape.Wecanhave numbers oftheroomsattachedto_keyswhich allowa

selection ofthe_{appropriate key}outof_manyother_placedonthe shelf.However,wecanalso consider

the_shapeof_key’sfeathermade of_{mechanically distinguishable}elementsor evenof molecules. In the

latter case, geometric structure ofthe _key is _carrying information. The two manifestations of

informationmakeoneoutof_verydifferent_{multiplicities,}but_theyarecloselyinterrelated.

The definition of information _presented above, which _{generalizes many} earlier _attempts and

which dueto_{its veryhigh}level of abstractioncanbe_appliedto_practicallyall instances of theuseof

theterm information,canbe usedto _develop amathematicalformalism for information.Itisnota

surprise,that theformalism is_{usingverygeneral}framework of_{algebra. [16]}

The_conceptof information_requiresavariety (many),whichcanbe understoodas anarbitraryset

\mathrm{S} _(called a carrier ofinformation). Information _system is this set \mathrm{S} _equippedwith the _family of subsets s\leftarrow _satisfying conditions: entire \mathrm{S} is in s\leftarrow, and together with every subfamily of s\leftarrow, its

intersection _belongsto s\leftarrow,i.e. s^{\infty} is aMoore family. Of course, this means that we have a closure

operator\mathrm{f}definedonS.TheMoore_family s\leftarrowof subsets is_simplythe_familyf‐C1 of all closedsubsets, i.e. subsetsA of\mathrm{S} suchthat _{\mathrm{A}=\mathrm{f}(\mathrm{A})}. Thefamily ofciosedsubsets s\leftarrow=\mathrm{f}-\mathrm{C}1 is equipped with the structure of a _{complete lattice \mathrm{L}_{\mathrm{f}} Uy} the set theoretical inclusion. \mathrm{L}_{\mathrm{f}} can _play a role of the

generalizationof_logicfornot_{necessarily linguistic}information_{systems,although}it doesnothaveto beaBooleanalgebra. In manycasesit maintainsallfundamental characteristicsofa_{logical system.}

[17]

Information itselfisadistinction ofasubset s_{0}^{\leftarrow}of s^{\infty},such that it is closed withrespectto(pair‐

wise) intersection and is _{dually‐hereditary,} i.e. with each subset _belongingto s_{0}^{\leftarrow}, all subsets of \mathrm{S}

includingit_belongtos_{0}^{\leftarrow}(i.e.\triangleleft^{\leftarrow}0 isafilter in \mathrm{L}_{\mathrm{f}}_).

The Moore _family s\leftarrow can_represent avariety ofstructures ofaparticular_{type (e.g. geometric,}

topological, algebraic, logical, etc.) definedon the subsets of S. This_corresponds tothe structural manifestation of inforrnation. Filter \mathrm{t}^{\leftarrow}0 in _{turn, in many} mathematical theories associated with localization,canbe usedasatool foridentification, i.e. selectionofanelement within the_{family s\leftrightarrow,}

and under some conditions in the set S. For _instance, in the context of Shannon’s selective information based on a probability distribution of the choice ofan element in \mathrm{S}, 30 consists of

elements in \mathrm{S}which have_probabilitymeasure 1,while\triangleleft^{\leftarrow}is_simplythesetofall subsets ofS.

The tools_developedinthe_precedingsectionallowustocharacterize s_{0}^{\leftarrow}intermsof its_symmetry. 6. Conclusion

The

_approach

_study

arbitrary

_presentation

_particular,

_special

_applications

_{already existing}

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